Estimating genotypes with independently sampled descent graphs

A method for estimating genotypic and identity-by-descent probabilities in complex pedigrees is described. The method consists of an algorithm for drawing independent genotype samples which are consistent with the pedigree and observed genotype. The probability distribution function for samples obtained using the algorithm can be evaluated up to a normalizing constant, and combined with the likelihood to produce a weight for each sample. Importance sampling is then used to estimate genotypic and identity-by-descent probabilities. On small but complex pedigrees, the genotypic probability estimates are demonstrated to be empirically unbiased. On large complex pedigrees, while the algorithm for obtaining genotype samples is feasible, importance sampling may require an infeasible number of samples to estimate genotypic probabilities with accuracy.     

A Model for Analysis of Population Structure

Arguments have been presented for the appropriateness of a multinomial Dirichlet distribution for describing single-locus genotypic frequencies in a subdivided population. This distribution is defined as a function of allele frequency, the average (over the entire population) inbreeding coefficient and the correlation between genotypes within a subdivision. Alternative parameterizations and their genetic interpretations are given.-We then show how information from a sample drawn from this subdivided population, in the absence of pedigrees, can be combined with the multinomial Dirichlet model to form a likelihood function. This likelihood function is then used as the basis for estimation and testing hypotheses concerning the genetic parameters of the model. Comparisons of this approach to the alternative procedure of Cockerham (1969) and (1973) are made using human data obtained from Tecumseh, Michigan and Monte Carlo simulations.-Finally, implications of these results to statistical inference and to mutation rates are presented.     

A comparison of alternative methods to compute conditional genotype probabilities for genetic evaluation with finite locus models

An increased availability of genotypes at marker loci has prompted the development of models that include the effect of individual genes. Selection based on these models is known as marker-assisted selection (MAS). MAS is known to be efficient especially for traits that have low heritability and non-additive gene action. BLUP methodology under non-additive gene action is not feasible for large inbred or crossbred pedigrees. It is easy to incorporate non-additive gene action in a finite locus model. Under such a model, the unobservable genotypic values can be predicted using the conditional mean of the genotypic values given the data. To compute this conditional mean, conditional genotype probabilities must be computed. In this study these probabilities were computed using iterative peeling, and three Markov chain Monte Carlo (MCMC) methods – scalar Gibbs, blocking Gibbs, and a sampler that combines the Elston Stewart algorithm with iterative peeling (ESIP). The performance of these four methods was assessed using simulated data. For pedigrees with loops, iterative peeling fails to provide accurate genotype probability estimates for some pedigree members. Also, computing time is exponentially related to the number of loci in the model. For MCMC methods, a linear relationship can be maintained by sampling genotypes one locus at a time. Out of the three MCMC methods considered, ESIP, performed the best while scalar Gibbs performed the worst.     

A study on the minimum number of loci required for genetic evaluation using a finite locus model

For a finite locus model, Markov chain Monte Carlo (MCMC) methods can be used to estimate the conditional mean of genotypic values given phenotypes, which is also known as the best predictor (BP). When computationally feasible, this type of genetic prediction provides an elegant solution to the problem of genetic evaluation under non-additive inheritance, especially for crossbred data. Successful application of MCMC methods for genetic evaluation using finite locus models depends, among other factors, on the number of loci assumed in the model. The effect of the assumed number of loci on evaluations obtained by BP was investigated using data simulated with about 100 loci. For several small pedigrees, genetic evaluations obtained by best linear prediction (BLP) were compared to genetic evaluations obtained by BP. For BLP evaluation, used here as the standard of comparison, only the first and second moments of the joint distribution of the genotypic and phenotypic values must be known. These moments were calculated from the gene frequencies and genotypic effects used in the simulation model. BP evaluation requires the complete distribution to be known. For each model used for BP evaluation, the gene frequencies and genotypic effects, which completely specify the required distribution, were derived such that the genotypic mean, the additive variance, and the dominance variance were the same as in the simulation model. For lowly heritable traits, evaluations obtained by BP under models with up to three loci closely matched the evaluations obtained by BLP for both purebred and crossbred data. For highly heritable traits, models with up to six loci were needed to match the evaluations obtained by BLP.     

An optimal algorithm for automatic genotype elimination

In an effort to accelerate likelihood computations on pedigrees, Lange and Goradia defined a genotype-elimination algorithm that aims to identify those genotypes that need not be considered during the likelihood computation. For pedigrees without loops, they showed that their algorithm was optimal, in the sense that it identified all genotypes that lead to a Mendelian inconsistency. Their algorithm, however, is not optimal for pedigrees with loops, which continue to pose daunting computational challenges. We present here a simple extension of the Lange-Goradia algorithm that we prove is optimal on pedigrees with loops, and we give examples of how our new algorithm can be used to detect genotyping errors. We also introduce a more efficient and faster algorithm for carrying out the fundamental step in the Lange-Goradia algorithm-namely, genotype elimination within a nuclear family. Finally, we improve a common algorithm for computing the likelihood of a pedigree with multiple loops. This algorithm breaks each loop by duplicating a person in that loop and then carrying out a separate likelihood calculation for each vector of possible genotypes of the loop breakers. This algorithm, however, does unnecessary computations when the loop-breaker vector is inconsistent. In this paper we present a new recursive loop breaker-elimination algorithm that solves this problem and illustrate its effectiveness on a pedigree with six loops.     

Selecting loop breakers in general pedigrees

The presence of loops in pedigrees poses severe computational problems in likelihood calculation that can be solved by creating an equivalent unlooped pedigree. We introduce a heuristic polynomial-time dynamic-programming algorithm, called SFH, that addresses the problem of selecting a minimal-cost set of loop breakers. We report computational experiments on simulated pedigrees with up to 1000 individuals and 361 loops, and multiple marriages. We compare the loop-breaker set selected by our method with that obtained using the software package FASTLINK 4.1P. Our approach outperforms FASTLINK 4.1P on the computational-time point of view, on the point of view of quality of the loop-breaker set obtained, and on the point of view of the size of the problem that can be addressed.     

Combining information within and between pedigrees for mapping complex traits.

This paper is concerned with efficient strategies for gene mapping using pedigrees containing small numbers of affecteds and identity-by-descent data from closely spaced markers throughout the genome. Particular attention is paid to additive traits involving phenocopies and/or locus heterogeneity. For a sample of pedigrees containing a particular configuration of affecteds, e.g., pairs of siblings together with a first cousin, we use a likelihood analysis to find 1-df statistics that are very efficient over a broad range of penetrances and allele frequencies. We identify configurations of affecteds that are particularly powerful for detecting linkage, and we show how pedigrees containing different numbers and configurations of affecteds can be efficiently combined in an overall test statistic.     

QTL analysis in arbitrary pedigrees with incomplete marker information

Mapping quantitative trait loci (QTL) in arbitrary outbred pedigrees is complicated by the combinatorial possibilities of allele flow relationships and of the founder allelic configurations. Exact methods are only available for rather short and simple pedigrees. Stochastic simulation using Markov chain Monte Carlo (MCMC) integration offers more flexibility. MCMC methods are less natural in a frequentist than in a Bayesian context, which we therefore adopt. Among the MCMC algorithms for updating marker locus genotypes, we implement the descent-graph algorithm. It can be used to update marker locus allele flow relationships and can handle arbitrarily complex pedigrees and missing marker information. Compared with updating marker genotypic information, updating QTL parameters, such as position, effects, and the allele flow relationships is relatively easy with MCMC. We treat the effect of each diploid combination of founder alleles as a random variable and only estimate the variance of these effects, ie, we model diploid genotypic effects instead of the usual partition in additive and dominance effects. This is a variant of the random model approach. The number of QTL alleles is generally unknown. In the Bayesian context, the number of QTL present on a linkage group can be treated as variable. Computer simulations suggest that the algorithm can indeed handle complex pedigrees and detect two QTL on a linkage group, but that the number of individuals in a single extended family is limited to about 50 to 100 individuals.     

An algorithm for automatic genotype elimination.

Automatic genotype elimination algorithms for a single locus play a central role in making likelihood computations on human pedigree data feasible. We present a simple algorithm that is fully efficient in pedigrees without loops. This algorithm can be easily coded and has been instrumental in greatly reducing computing times for pedigree analysis. A contrived counter-example demonstrates that some superfluous genotypes cannot be excluded for inbred pedigrees.     

Improving linkage analysis in outcrossed forest trees - an example from Acacia mangium

Mapping in forest trees generally relies on outbred pedigrees in which genetic segregation is the result of meiotic recombination from both parents. The currently available mapping packages are not optimal for outcrossed pedigrees as they either cannot order phase-ambiguous data or only use pairwise information when ordering loci within linkage groups. A new package, OUTMAP, has been developed for mapping codominant loci in outcrossed trees. A comparison of maps produced using linkage data from two pedigrees of Acacia mangium Willd demonstrated that the marker orders produced using OUTMAP were consistently of higher likelihood than those produced by JOINMAP. In addition, the maps were produced more efficiently, without the need for recoding data or the detailed investigation of pairwise recombination fractions which was necessary to select the optimal marker order using JOINMAP. Distances between markers often varied from those calculated by JOINMAP, resulting in an increase in the estimated genome length. OUTMAP can be used with all segregation types to determine phase and to calculate the likelihood of alternative marker orders, with a choice of three optimisation methods.     

Linkage Analysis with an Alternative Formulation for the Mixed Model of Inheritance: The Finite Polygenic Mixed Model

This paper presents an extension of the finite polygenic mixed model of FERNANDO et al. (1994) to linkage analysis. The finite polygenic mixed model, extended for linkage analysis, leads to a likelihood that can be calculated using efficient algorithms developed for oligogenic models. For comparison, linkage analysis of 5 simulated 4021-member pedigrees was performed using the usual mixed model of inheritance, approximated by HASSTEDT (1982), and the finite polygenic mixed model extended for linkage analysis presented here. Maximum likelihood estimates of the finite polygenic mixed model could be inferred to be closer to the simulated values in these pedigrees.     

An algorithm for sampling descent graphs in large complex pedigrees efficiently

No exact method for determining genotypic and identity-by-descent probabilities is available for large complex pedigrees. Approximate methods for such pedigrees cannot be guaranteed to be unbiased. A new method is proposed that uses the Metropolis-Hastings algorithm to sample a Markov chain of descent graphs which fit the pedigree and known genotypes. Unknown genotypes are determined from each descent graph. Genotypic probabilities are estimated as their means. The algorithm is shown to be unbiased for small complex pedigrees and feasible and consistent for moderately large complex pedigrees.     

Comparisons of different sampling designs for the determination of genetic transmission mechanisms in quantitative traits

The power provided by several sampling designs to detect segregation at a major locus was investigated in a simulation study using phenotypes constructed from a major-locus genotypic mean, a background polygenic effect, and an individual-specific environmental effect. Questions of which relatives, how many relatives, and how many independent pedigrees to collect were considered, using configurations ranging from nuclear families of size 5 to 4-generation pedigrees of size 45. Each configuration contained a single proband whose phenotype exceeded the 95th percentile in a population where 2.5% carry the disease susceptibility allele. Results suggest that, under the conditions simulated, when total sample size is fixed, samples composed of 3-generation pedigrees of intermediate size provide a greater magnitude of support for the presence of a major locus than do samples composed of nuclear families or 4-generation pedigrees. This study is the first to consider both the discriminatory power and estimation efficiency in comparing alternative sampling strategies for pedigree data.     

Maximum likelihood haplotyping for general pedigrees

Haplotype data is valuable in mapping disease-susceptibility genes in the study of Mendelian and complex diseases. We present algorithms for inferring a most likely haplotype configuration for general pedigrees, implemented in the newest version of the genetic linkage analysis system SUPERLINK. In SUPERLINK, genetic linkage analysis problems are represented internally using Bayesian networks. The use of Bayesian networks enables efficient maximum likelihood haplotyping for more complex pedigrees than was previously possible. Furthermore, to support efficient haplotyping for larger pedigrees, we have also incorporated a novel algorithm for determining a better elimination order for the variables of the Bayesian network. The presented optimization algorithm also improves likelihood computations. We present experimental results for the new algorithms on a variety of real and semiartificial data sets, and use our software to evaluate MCMC approximations for haplotyping.     

Optimizing exact genetic linkage computations.

Genetic linkage analysis is a challenging application which requires Bayesian networks consisting of thousands of vertices. Consequently, computing the probability of data, which is needed for learning linkage parameters, using exact computation procedures calls for an extremely efficient implementation that carefully optimizes the order of conditioning and summation operations. In this paper, we present the use of stochastic greedy algorithms for optimizing this order. Our algorithm has been incorporated into the newest version of SUPERLINK, which is a fast genetic linkage program for exact likelihood computations in general pedigrees. We demonstrate an order of magnitude improvement in run times of likelihood computations using our new optimization algorithm and hence enlarge the class of problems that can be handled effectively by exact computations.     

Segregation analysis of continuous phenotypes by using higher sample moments.

The present article discusses the use of computational methods based on generalized estimating equations (GEE), as a potential alternative to full maximum-likelihood methods, for performing segregation analysis of continuous phenotypes by using randomly selected family data. The method that we propose can estimate effect and degree of dominance of a major gene in the presence of additional nongenetic or polygenetic familial associations, by relating sample moments to their expectations calculated under the genetic model. It is known that all parameters in basic major-gene models cannot be identified, for estimation purposes, solely in terms of the first two sample moments of data from randomly selected families. Thus, we propose the use of higher (third order) sample moments to resolve this identifiability problem, in a pseudo-profile likelihood estimation scheme. In principle, our methods may be applied to fitting genetic models by using complex pedigrees and for estimation in the presence of missing phenotype data for family members. In order to assess its statistical efficiency we compare several variants of the method with each other and with maximum-likelihood estimates provided by the SAGE computer package in a simulation study.     

Estimation and visualization of identity-by-descent within pedigrees simplifies interpretation of complex trait analysis

Linkage analysis identifies markers that appear to be co-inherited with a trait within pedigrees. The inheritance of a chromosomal segment may be probabilistically reconstructed, with missing data complicating inference. Inheritance patterns are further obscured in the analysis of complex traits, where variants in one or more genes may contribute to phenotypic variation within a pedigree. In this case, determining which relatives share a trait variant is not simple. We describe how to represent these patterns of inheritance for marker loci. We summarize how to sample patterns of inheritance consistent with genotypic and pedigree data using gl_auto, available in MORGAN v3.0. We describe identification of classes of equivalent inheritance patterns with the program IBDgraph. We finally provide an example of how these programs may be used to simplify interpretation of linkage analysis of complex traits in general pedigrees. We borrow information across loci in a parametric linkage analysis of a large pedigree. We explore the contribution of each equivalence class to a linkage signal, illustrate estimated patterns of identity-by-descent sharing, and identify a haplotype tagging the chromosomal segment driving the linkage signal. Haplotype carriers are more likely to share the linked trait variant, and can be prioritized for subsequent DNA sequencing.     

COLONY: a program for parentage and sibship inference from multilocus genotype data

Pedigrees, depicting genealogical relationships between individuals, are important in several research areas. Molecular markers allow inference of pedigrees in wild species where relationship information is impossible to collect by observation. Marker data are analysed statistically using methods based on Mendelian inheritance rules. There are numerous computer programs available to conduct pedigree analysis, but most software is inflexible, both in terms of assumptions and data requirements. Most methods only accommodate monogamous diploid species using codominant markers without genotyping error. In addition, most commonly used methods use pairwise comparisons rather than a full-pedigree likelihood approach, which considers the likelihood of the entire pedigree structure and allows the simultaneous inference of parentage and sibship. Here, we describe colony, a computer program implementing full-pedigree likelihood methods to simultaneously infer sibship and parentage among individuals using multilocus genotype data. colony can be used for both diploid and haplodiploid species; it can use dominant and codominant markers, and can accommodate, and estimate, genotyping error at each locus. In addition, colony can carry out these inferences for both monoecious and dioecious species. The program is available as a Microsoft Windows version, which includes a graphical user interface, and a Macintosh version, which uses an R-based interface.     

Estimating ethnic admixture from pedigree data

This paper introduces a likelihood method of estimating ethnic admixture that uses individuals, pedigrees, or a combination of individuals and pedigrees. For each founder of a pedigree, admixture proportions are calculated by conditioning on the pedigree-wide genotypes at all ancestry-informative markers. These estimates are then propagated down the pedigree to the nonfounders by a simple averaging process. The large-sample standard errors of the founders' proportions can be similarly transformed into standard errors for the admixture proportions of the descendants. These standard errors are smaller than the corresponding standard errors when each individual is treated independently. Both hard and soft information on a founder's ancestry can be accommodated in this scheme, which has been implemented in the genetic software package Mendel. The utility of the method is demonstrated on simulated data and a real data example involving Mexican families of mixed Amerindian and Spanish ancestry.